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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 348075.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 348075.z1 | 348075z5 | \([1, -1, 1, -9130280, 10613457722]\) | \(7389727131216686257/6115533215337\) | \(69659745530948015625\) | \([2]\) | \(12582912\) | \(2.7359\) | |
| 348075.z2 | 348075z3 | \([1, -1, 1, -696155, 87669722]\) | \(3275619238041697/1605271262049\) | \(18285042969276890625\) | \([2, 2]\) | \(6291456\) | \(2.3893\) | |
| 348075.z3 | 348075z2 | \([1, -1, 1, -371030, -85947028]\) | \(495909170514577/6224736609\) | \(70903640436890625\) | \([2, 2]\) | \(3145728\) | \(2.0428\) | |
| 348075.z4 | 348075z1 | \([1, -1, 1, -369905, -86500528]\) | \(491411892194497/78897\) | \(898686140625\) | \([2]\) | \(1572864\) | \(1.6962\) | \(\Gamma_0(N)\)-optimal |
| 348075.z5 | 348075z4 | \([1, -1, 1, -63905, -224153278]\) | \(-2533811507137/1904381781393\) | \(-21692098728679640625\) | \([2]\) | \(6291456\) | \(2.3893\) | |
| 348075.z6 | 348075z6 | \([1, -1, 1, 2535970, 669452222]\) | \(158346567380527343/108665074944153\) | \(-1237763119285742765625\) | \([2]\) | \(12582912\) | \(2.7359\) |
Rank
sage: E.rank()
The elliptic curves in class 348075.z have rank \(2\).
Complex multiplication
The elliptic curves in class 348075.z do not have complex multiplication.Modular form 348075.2.a.z
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
